CFD-based scale-up of hydrodynamics and mixing in bubble columns

Unsteady and three-dimensional Eulerian–Eulerian CFD simulations of bubble column reactors under operating conditions of industrial interest are discussed in this work. The flow pattern in this equipment depends strongly on the interactions between the gas and liquid phases, mainly via the drag force. In this work, a correlation for the drag force coefficient is tested and improved to consider the so-called swarm effect that modifies the drag force at high gas volume fractions. The improved swarm factor proposed in this work is the adjustment of the swarm factor proposed by Simonnet et al. (2008). This new swarm factor is suitable for very high gas volume fractions without generating stability problems, which were encountered with the original formulation. It delivers an accurate prediction of gas volume fraction and liquid velocity in a wide range of tested operating conditions. Results are validated by comparison with experimental data on bubble column reactors at different scales and for several operating conditions. Hydrodynamics is well predicted for every operating condition at different scales. Several turbulence models are tested. Finally, the contribution of Bubble Induced Turbulence (BIT), as proposed by Alméras et al. (2015), on mixing is evaluated via an analysis of the mixing time

Study of impurities in spin-Peierls systems including lattice relaxation

The effects of magnetic and non-magnetic impurities in spin-Peierls systems are investigated allowing for lattice relaxation and quantum fluctuations. We show that, in isolated chains, strong bonds form next to impurities, leading to the appearance of magneto-elastic solitons. Generically, these solitonic excitations do not bind to impurities. However, interchain elastic coupling produces an attractive potential at the impurity site which can lead to the formation of bound states. In addition, we predict that small enough chain segments do not carry magnetic moments at the ends

Phase diagram of a Heisenberg spin-Peierls model with quantum phonons

Using a new version of the density-matrix renormalization group we determine the phase diagram of a model of an antiferromagnetic Heisenberg spin chain where the spins interact with quantum phonons. A quantum phase transition from a gapless spin-fluid state to a gapped dimerized phase occurs at a non-zero value of the spin-phonon coupling. The transition is in the same universality class as that of a frustrated spin chain, which the model maps to in the anti-adiabatic limit. We argue that realistic modeling of known spin-Peierls materials should include the effects of quantum phonons.Comment: RevTeX, 5 pages, 3 eps figures included using epsf. Improved theories in adiabatic and non-adiabatic regimes give better agreement with DMRG. This version accepted in Physical Review Letter

Antiferromagnetism in doped anisotropic two-dimensional spin-Peierls systems

We study the formation of antiferromagnetic correlations induced by impurity doping in anisotropic two-dimensional spin-Peierls systems. Using a mean-field approximation to deal with the inter-chain magnetic coupling, the intra-chain correlations are treated exactly by numerical techniques. The magnetic coupling between impurities is computed for both adiabatic and dynamical lattices and is shown to have an alternating sign as a function of the impurity-impurity distance, hence suppressing magnetic frustration. An effective model based on our numerical results supports the coexistence of antiferromagnetism and dimerization in this system.Comment: 5 pages, 4 figures; final version to appear in Phys. Rev.

Peierls Dimerization with Non-Adiabatic Spin-Phonon Coupling

We study the magnetic properties of a frustrated Heisenberg spin chain with a dynamic spin-phonon interaction. By Lanczos diagonalization, preserving the full lattice dynamics, we explore the non-adiabatic regime with phonon frequencies comparable to the exchange coupling energy which is e.g. the relevant limit for the spin-Peierls compound $CuGeO_3$. When compared to the static limit of an alternating spin chain the magnetic properties are strongly renormalized due to the coupled dynamics of spin and lattice degrees of freedom. The magnitude of the spin triplet excitation gap changes from a strong to a weak dimerization dependence with increasing phonon frequencies implying the necessity to include dynamic effects in an attempt for a quantitative description of the spin-Peierls state.Comment: 4 pages, 5 figure

Excitation Spectra of Structurally Dimerized and Spin-Peierls Chains in a Magnetic Field

The dynamical spin structure factor and the Raman response are calculated for structurally dimerized and spin-Peierls chains in a magnetic field, using exact diagonalization techniques. In both cases there is a spin liquid phase composed of interacting singlet dimers at small fields h < h_c1, an incommensurate regime (h_c1 < h < h_c2) in which the modulation of the triplet excitation spectra adapts to the applied field, and a fully spin polarized phase above an upper critical field h_c2. For structurally dimerized chains, the spin gap closes in the incommensurate phase, whereas spin-Peierls chains remain gapped. In the spin liquid regimes, the dominant feature of the triplet spectra is a one-magnon bound state, separated from a continuum of states at higher energies. There are also indications of a singlet bound state above the one-magnon triplet.Comment: RevTex, 10 pages with 8 eps figure

Temperature Dependence of Spin and Bond Ordering in a Spin-Peierls System

We investigate thermodynamic properties of a one-dimensional S=1/2 antiferromagnetic Heisenberg model coupled to a lattice distortion by a quantum Monte Carlo method. In particular we study how spin and lattice dimerize as a function of the temperature, which gives a fundamental process of the spin-Peierls transition in higher dimensions. The degree of freedom of the lattice is taken into account adiabatically and the thermal distribution of the lattice distortion is obtained by the thermal bath algorithm. We find that the dimerization develops as the temperature decreases and it converges to the value of the dimerization of the ground state at T=0. Furthermore we find that the coupling constants of spins fluctuate quite largly at high temperature and there thermodynamic properties deviate from those of the uniform chain. Doping of non-magnetic impurities causes cut of the chain into short chains with open boundary. We investigate thermodynamic properties of open chains taking relaxation of the lattice into consideration. We find that strong bonds locate at the edges and a defect of the bond alternation appears in the chain with odd number of sites, which causes enhancement of the staggered magnetic order. We find a spreaded staggered structure which indicates that the defect moves diffusively in the chain even at very low temperature.Comment: 7 pages, 17 figures; added comments on section 2 and 3, corrected typo

Study of spin-Peierls transition in alpha'--NaV2O5 by infrared reflectivity

Polarized infrared reflectivity measurements have been performed on single crystals of the spin-Peierls compound alpha'-NaV2O5 in the temperature range 20-300 K. Pronounced spectral features associated with the formation of the dimerized phase were detected both in the a- and b-polarizations (perpendicular and parallel to the spin-1/2 chains, respectively). The temperature dependence of a salient spectral line at 718 cm^-1 sharply rising below the transition temperature T_SP obeys a (1-T/T_SP)^(2beta) law with T_SP \simeq 34.3$K and beta \simeq 0.25. In addition, a continuum signal is observed in the whole temperature range in the a-polarized optical conductivity spectra. In order to interpret these results, calculations of the static dimerization and of the optical conductivity based on a mean-field and a dynamical treatment of the lattice respectively are proposed.Comment: 4 pages, 5 PostScript figures, Minor change

Three-Dimensional Ordering in Weakly Coupled Antiferromagnetic Ladders and Chains

A theoretical description is presented for low-temperature magnetic-field induced three-dimensional (3D) ordering transitions in strongly anisotropic quantum antiferromagnets, consisting of weakly coupled antiferromagnetic spin-1/2 chains and ladders. First, effective continuum field theories are derived for the one-dimensional subsystems. Then the Luttinger parameters, which determine the low-temperature susceptibilities of the chains and ladders, are calculated from the Bethe ansatz solution for these effective models. The 3D ordering transition line is obtained using a random phase approximation for the weak inter-chain (inter-ladder) coupling. Finally, considering a Ginzburg criterion, the fluctuation corrections to this approach are shown to be small. The nature of the 3D ordered phase resembles a Bose condensate of integer-spin magnons. It is proposed that for systems with higher spin degrees of freedom, e.g. N-leg spin-1/2 ladders, multi-component condensates can occur at high magnetic fields.Comment: RevTex, 18 pages with 7 figure

Dynamical properties of the spin-Peierls compound \alpha'--NaV2O5

Dynamical properties of the novel inorganic spin-Peierls compound \alpha'--NaV2O5 are investigated using a one-dimensional dimerized Heisenberg model. By exact diagonalizations of chains with up to 28 sites, supplemented by a finite-size scaling analysis, the dimerization parameter \delta is determined by requiring that the model reproduces the experimentally observed spin gap \Delta. The dynamical and static spin structure factors are calculated. As for CuGeO3, the existence of a low energy magnon branch separated from the continuum is predicted. The present calculations also suggest that a large magnetic Raman scattering intensity should appear above an energy threshold of 1.9 \Delta. The predicted photoemission spectrum is qualitatively similar to results for an undimerized chain due to the presence of sizable short-range antiferromagnetic correlations.Comment: 4 pages, latex, minor misprints corrected and a few references adde